Azagra Rueda, Daniel and Dobrowolski, Tadeusz (1998) Smooth negligibility of compact sets in infinitedimensional Banach spaces, with applications. Mathematische Annalen, 312 (3). pp. 445463. ISSN 00255831

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Abstract
This article deals with smooth removability of compact sets in infinitedimensional Banach spaces. The main result states that ifX is an infinitedimensional Banach space which has a not necessarily equivalent Cpsmooth norm and K is a compact subset of X, then X and X r K are Cp diffeomorphic.
The proof relies on the construction of a “deleting path” through a nontrivial refinement of Bessaga’s incompletenorm technique. However, norms are not at present available and the construction requires the use of asymmetric functionals. The noncompleteness of such functionals relies in turn on James’ theorem on existence of linear functionals which do not attain their norm on every nonreflexive space. Applications are given which show that several important theorems on finitedimensional spaces completely fail in the infinitedimensional case: for instance, on any Banach space isomorphic to its Cartesian square and for any natural number n _ 2 there exists a C1diffeomorphism of pure period n with no fixed point. This work opens the way to several interesting open questions on nonseparable Banach spaces: Does every Banach space with a C1 smooth norm admit a nonequivalent C1smooth norm? In which Banach spaces is every compact subset the set where a certain C1 realvalued function vanishes?
Item Type:  Article 

Uncontrolled Keywords:  Cp diffeomorphisms; Cp smooth norm; Complete smooth classification of the convex bodies of every Banach space; Garay’s phenomena for ODE’s in Banach spaces; Existence of periodic diffeomorphisms without fixed points 
Subjects:  Sciences > Mathematics > Functional analysis and Operator theory 
ID Code:  13947 
Deposited On:  23 Nov 2011 12:32 
Last Modified:  06 Feb 2014 09:56 
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